Question

Find the Cartesian equation of the plane which contains the line L1: x=1-3t, y=-2+t, t in R, and z=4-t and is parallel to the line.

Answer 1

To find the Cartesian equation of the plane that contains a given line L and is parallel to it, we need to find a point on the line and a normal vector to the plane. The Cartesian equation of the plane is -3x + y - z + 9 = 0.

Step-by-step explanation:

To find the Cartesian equation of the plane that contains the given line and is parallel to it, we need to find a point on the plane and a normal vector to the plane. Since the plane is parallel to the line, the direction vector of the line will be normal to the plane.

The direction vector of the line is v = (-3, 1, -1). Now, let's choose a point on the line, say when t = 0. Plugging in the values, we get P(1, -2, 4).
So, the Cartesian equation of the plane is -3(x-1) + (y+2) - (z-4) = 0, which simplifies to -3x + y - z + 9 = 0.